Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
HermiteH






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HermiteH[nu,z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving exponential function and a power function





http://functions.wolfram.com/07.01.21.0008.01









  


  










Input Form





Integrate[(z^(\[Alpha] - 1) HermiteH[\[Nu], z])/E^(p z^2), z] == 2^\[Nu] Sqrt[Pi] ((-(z^\[Alpha]/(2 Gamma[(1 - \[Nu])/2] (p z^2)^(\[Alpha]/2)))) Sum[(Pochhammer[-(\[Nu]/2), k]/(Pochhammer[1/2, k] k! p^k)) Gamma[\[Alpha]/2 + k, p z^2], {k, 0, Infinity}] + (z^(1 + \[Alpha])/(Gamma[-(\[Nu]/2)] (p z^2)^((\[Alpha] + 1)/2))) Sum[(Pochhammer[(1 - \[Nu])/2, k]/(Pochhammer[3/2, k] k! p^k)) Gamma[(\[Alpha] + 1)/2 + k, p z^2], {k, 0, Infinity}])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "p"]], " ", SuperscriptBox["z", "2"]]]], RowBox[List["HermiteH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["z", "\[Alpha]"], RowBox[List["2", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["p", " ", SuperscriptBox["z", "2"]]], ")"]], FractionBox["\[Alpha]", "2"]]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], SuperscriptBox["p", "k"]]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["\[Alpha]", "2"], " ", "+", "k"]], ",", RowBox[List["p", " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]]]], "+", RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["1", "+", "\[Alpha]"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["p", " ", SuperscriptBox["z", "2"]]], ")"]], FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], SuperscriptBox["p", "k"]]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], " ", "+", "k"]], ",", RowBox[List["p", " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]]]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> p </mi> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mn> 2 </mn> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <msup> <mi> z </mi> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;\[Nu]&quot;]], &quot;2&quot;], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mi> p </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;3&quot;, &quot;2&quot;], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> p </mi> <mi> k </mi> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <msup> <mi> z </mi> <mi> &#945; </mi> </msup> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#945; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, FractionBox[&quot;\[Nu]&quot;, &quot;2&quot;]]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> &#945; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mi> p </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> p </mi> <mi> k </mi> </msup> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> HermiteH </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <cn type='rational'> 3 <sep /> 2 </cn> <ci> k </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> <apply> <power /> <ci> p </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> &#945; </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> k </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> <apply> <power /> <ci> p </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "p_"]], " ", SuperscriptBox["z_", "2"]]]], " ", RowBox[List["HermiteH", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["\[Alpha]", "2"], "+", "k"]], ",", RowBox[List["p", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", SuperscriptBox["p", "k"]]]]]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["\[Alpha]", "/", "2"]]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["1", "+", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], "+", "k"]], ",", RowBox[List["p", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", SuperscriptBox["p", "k"]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", " ", SuperscriptBox["z", "2"]]], ")"]], FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"]]]]]]], ")"]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29





© 1998- Wolfram Research, Inc.